#### The angle between the straight lines, whose direction cosines are given by the equations , is :Option: 1Option: 2Option: 3Option: 4

$2l+2m-n= 0\\$

$\Rightarrow n= 2l+2m\\$            ...............(1)

$mn+nl+lm= 0\\$

$\Rightarrow n\left ( m+l \right )+lm= 0\\$

$\Rightarrow 2\left ( l+m \right )\left ( l+m \right )+lm= 0\\$

$\Rightarrow 2l^{2}+4ml+2m^{2}+lm= 0\\$

$\Rightarrow 2l\left ( l+2m \right )+m\left ( l+2m \right )= 0\\$

$\Rightarrow \left ( 2l+m \right )\left ( l+2m \right )= 0\\$

$\Rightarrow m= -2l\: or\: l= -2m\\$               ..........(2)

From (1) and (2)

$n= 2l+2m\\$

$= 2l-4l= -2l\: or\: -4m+2m= -2m\\$

So $m= -2l= n\:\: OR\: \: l= -2m= n\\$

Also $l^{2}+m^{2}+n^{2}= 1\\$

$\Rightarrow l^{2}+4l^{2}+4l^{2}-1\Rightarrow l= \pm \frac{1}{3}\\$

$OR\: \: 4m^{2}+m^{2}+4m^{2}= 1\Rightarrow m= \frac{1}{3}\\$

So $L_{1}:\pm \left ( \frac{1}{3} ,\frac{-2}{3},\frac{-2}{3}\right )\: L_{2}:\pm \left ( \frac{-2}{3},\frac{1}{3}, \frac{-2}{3}\right )\\$

Angle $\cos\theta= \left | l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2} \right |\\$

$= \left | \frac{-2}{9}-\frac{2}{9}+\frac{4}{9} \right |= 0\\$

$\Rightarrow \theta= 90^{\circ}= \frac{\pi}{2}$